Pure square root finding

10

1. The complex number \(u\) is given by \(u = - 3 - ( 2 \sqrt { } 10 )\) i. Showing all necessary working and without using a calculator, find the square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
2. On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of the complex number \(z\).
   [0pt] [5] If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The complex number \(u\) is given by \(u = 10 - 4 \sqrt { 6 } \mathrm { i }\).
Find the two square roots of \(u\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.

2 Use an algebraic method to find the square roots of the complex number \(15 + 8 \mathrm { i }\).

3 Use an algebraic method to find the square roots of \(11 + ( 12 \sqrt { 5 } ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.

3 Use an algebraic method to find the square roots of \(3 + ( 6 \sqrt { 2 } )\) i. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.

1 In this question you must show detailed reasoning.
Find the square roots of \(24 + 10 \mathrm { i }\), giving your answers in the form \(a + b \mathrm { i }\).

1 In this question you must show detailed reasoning.
Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).

1 Without using a calculator, determine the possible values of \(a\) and \(b\) for which \(( a + \mathrm { i } b ) ^ { 2 } = 21 - 20 \mathrm { i }\).

The square roots of \(24 - 7i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7i\) in exact Cartesian form. [5]

The square roots of \(6 - 8i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8i\) in exact Cartesian form. [5]

The complex number \(z = a + ib\), where \(a\) and \(b\) are real numbers, satisfies the equation $$z^2 + 16 - 30i = 0.$$

1. Show that \(ab = 15\). [2]
2. Write down a second equation in \(a\) and \(b\) and hence find the roots of \(z^2 + 16 - 30i = 0\). [4]

Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]

Use an algebraic method to find the square roots of the complex number \(21 - 20\text{i}\). [6]

In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(-77 - 36\text{i}\). [6]